1. Field of the Invention
The present invention relates to performing arithmetic operations on interval operands within a computer system. More specifically, the present invention relates to a method and an apparatus for using a computer system to solve a system of nonlinear equations with interval arithmetic.
2. Related Art
Rapid advances in computing technology make it possible to perform trillions of computational operations each second. This tremendous computational speed makes it practical to perform computationally intensive tasks as diverse as predicting the weather and optimizing the design of an aircraft engine. Such computational tasks are typically performed using machine-representable floating-point numbers to approximate values of real numbers. (For example, see the Institute of Electrical and Electronics Engineers (IEEE) standard 754 for binary floating-point numbers.)
In spite of their limitations, floating-point numbers are generally used to perform most computational tasks.
One limitation is that machine-representable floating-point numbers have a fixed-size word length, which limits their accuracy. Note that a floating-point number is typically encoded using a 32, 64 or 128-bit binary number, which means that there are only 232, 264 or 2128 possible symbols that can be used to specify a floating-point number. Hence, most real number values can only be approximated with a corresponding floating-point number. This creates estimation errors that can be magnified through even a few computations, thereby adversely affecting the accuracy of a computation.
A related limitation is that floating-point numbers contain no information about their accuracy. Most measured data values include some amount of error that arises from the measurement process itself. This error can often be quantified as an accuracy parameter, which can subsequently be used to determine the accuracy of a computation. However, floating-point numbers are not designed to keep track of accuracy information, whether from input data measurement errors or machine rounding errors. Hence, it is not possible to determine the accuracy of a computation by merely examining the floating-point number that results from the computation.
Interval arithmetic has been developed to solve the above-described problems. Interval arithmetic represents numbers as intervals specified by a first (left) endpoint and a second (right) endpoint. For example, the interval [a, b], where a<b, is a closed, bounded subset of the real numbers, R, which includes a and b as well as all real numbers between a and b. Arithmetic operations on interval operands (interval arithmetic) are defined so that interval results always contain the entire set of possible values. The result is a mathematical system for rigorously bounding numerical errors from all sources, including measurement data errors, machine rounding errors and their interactions. (Note that the first endpoint normally contains the “infimum”, which is the largest number that is less than or equal to each of a given set of real numbers. Similarly, the second endpoint normally contains the “supremum”, which is the smallest number that is greater than or equal to each of the given set of real numbers.)
One commonly performed computational operation is to find the roots of a nonlinear equation. This can be accomplished using Newton's method. The interval version of Newton's method works in the following manner. From the mean value theorem,ƒ(x)−ƒ(x*)=(x−x*)ƒ′(ξ),where ξ is some generally unknown point between x and x*. If x* is a zero of ƒ, then ƒ(x*)=0 and, from the previous equation,x*=x−ƒ(x)/ƒ′(ξ).Let X be an interval containing both x and x*. Since ξ is between x and x*, it follows that ξ∈X Moreover, from basic properties of interval analysis it follows that ƒ′(ξ)∈ƒ′(X). Hence, x*∈N(x,X) whereN(x,X)=x−ƒ(x)/ƒ′(X).Temporarily assume 0∉ƒ′(X) so that N(x,X) is a finite interval. Since any zero of ƒ in X is also in N(x,X), the zero is in the intersection X∩N(x,X). Using this fact, we define an algorithm for finding x*. Let X0 be an interval containing x*. For n=0, 1, 2, . . . definexn=m(Xn)N(xn,Xn)=xn−ƒI(xn)/ƒ′(Xn)Xn+1=Xn∩N(xn,Xn),wherein m(X) is the midpoint of the interval X, and the notation ƒI(xn) makes explicit the fact that evaluating ƒ at the point xn must be done using interval arithmetic. We call xn the point of expansion for the Newton method. It is not necessary to choose xn to be the midpoint of Xn. The only requirement is that xn∈Xn to assure that x*∈N(xn, Xn). However, it is convenient and efficient to choose xn=m(Xn). Note that the roots of an interval equation can be intervals rather than points when the equation contains non-degenerate interval constants or parameters.
The interval version of Newton's algorithm for finding roots of nonlinear equations is designed to work best “in the small” when nonlinear equations are approximately linear. For large intervals containing multiple roots, the interval Newton algorithm splits the given interval into two sub-intervals that are then processed independently. By this mechanism all the roots of a nonlinear equation can be found.
One problem is applying the multivariate generalization of the interval Newton algorithm to large n-dimensional interval vectors (or boxes) that contain multiple roots. In this case, the process of splitting in n-dimensions can lead to exponential growth in the number of boxes to process.
It is well known that this problem (and even the problem of computing “sharp” bounds on the range of a function of n-variables over an n-dimensional box) is an “NP-hard” problem. In general, NP-hard problems require an exponentially increasing amount of work to solve as n, the number of independent variables, increases.
Because NP-hardness is a worst-case property and because many practical engineering and scientific problems have relatively simple structure, one problem is to use this simple structure of real problems to improve the efficiency of interval nonlinear equation solving algorithms.
Hence, what is needed is a method and an apparatus for using the structure of nonlinear equations to improve the efficiency of interval root-bounding software. To this end, what is needed is a method and apparatus that efficiently deletes boxes or parts of “large” boxes that the interval Newton algorithm can only split.